Because I'm primarily interested in this question from the point of view of $\infty$-categories (in this case, modeled by quasicategories), I'll ask this question using that terminology. In particular, I'll just say "category" instead of $\infty$-category or quasicategory, to keep things legible. I suspect that, up to adding in the word "homotopy," a lot of what I'm going to say holds very generally.

It's not hard to work out that if $Top$ is the category of $\infty$-groupoids (or anima, according to Peter Scholze), then there is an equivalence between comodules over any $X\in Top$, with respect to the Cartesian monoidal structure, and the slice category $Top/X$. I'll write $Comod_X(Top)\simeq Top/X$. Furthermore, by Lurie's straightening/unstraightening (a.k.a. the $\infty$-categorical Grothendieck construction), we have an equivalence $Top/X\simeq Fun(X,Top)$. The first category, $Comod_X(Top)$, is clearly comonadic over $Top$ via the forgetful functor $U\colon Comod_X(Top)→Top$. Moreover, the colimit functor $Fun(X,Top)→Top$ factors through $Comod_X(Top)$ and is the left adjoint of an adjoint equivalence realizing the composite equivalence $Comod_X(Top)?Fun(X,Top)$. Thus, $Fun(X,Top)$ is comonadic over $Top$ via the colimit functor $colim\colon Fun(X,Top)→Top$.

My question is whether or not this is more generally true, perhaps for some generic abstract reasons that I'm not aware or, or not seeing. Note that it is *not* in general true (I don't think) that $colim:Fun(D,C)→C$ is comonadic for arbitrary categories $D$ and $C$. However, I'm happy with assuming that $D$ is an $∞$-groupoid and that $C$ is "nice," i.e. at least (locally) presentable.

So really the question is, are there check-able conditions under which the colimit functor $colim:Fun(X,C)→C$ is comonadic, under the assumptions that $X$ is an $∞$-groupoid and $C$ is presentable?